| L(s) = 1 | + 0.347·2-s − 1.87·4-s − 0.652·5-s − 1.34·8-s − 0.226·10-s − 0.532·11-s + 13-s + 3.29·16-s − 1.12·17-s + 0.305·19-s + 1.22·20-s − 0.184·22-s + 8.00·23-s − 4.57·25-s + 0.347·26-s − 7.78·29-s + 0.588·31-s + 3.83·32-s − 0.389·34-s − 2.87·37-s + 0.106·38-s + 0.879·40-s + 1.04·41-s + 2.94·43-s + 44-s + 2.78·46-s + 4.12·47-s + ⋯ |
| L(s) = 1 | + 0.245·2-s − 0.939·4-s − 0.291·5-s − 0.476·8-s − 0.0716·10-s − 0.160·11-s + 0.277·13-s + 0.822·16-s − 0.271·17-s + 0.0700·19-s + 0.274·20-s − 0.0393·22-s + 1.66·23-s − 0.914·25-s + 0.0681·26-s − 1.44·29-s + 0.105·31-s + 0.678·32-s − 0.0667·34-s − 0.473·37-s + 0.0172·38-s + 0.139·40-s + 0.162·41-s + 0.448·43-s + 0.150·44-s + 0.410·46-s + 0.602·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 5 | \( 1 + 0.652T + 5T^{2} \) |
| 11 | \( 1 + 0.532T + 11T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 - 0.305T + 19T^{2} \) |
| 23 | \( 1 - 8.00T + 23T^{2} \) |
| 29 | \( 1 + 7.78T + 29T^{2} \) |
| 31 | \( 1 - 0.588T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 - 2.94T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81142875546424320435383079315, −7.14050289053145095347751745227, −6.22655112726017489396741161481, −5.43267016678994048285916373747, −4.91508510671321101046514596845, −3.98925371599172244956052773113, −3.51194958299602063691156481868, −2.47420384923129292288837054175, −1.17515817430605305739623155832, 0,
1.17515817430605305739623155832, 2.47420384923129292288837054175, 3.51194958299602063691156481868, 3.98925371599172244956052773113, 4.91508510671321101046514596845, 5.43267016678994048285916373747, 6.22655112726017489396741161481, 7.14050289053145095347751745227, 7.81142875546424320435383079315
